On an Erd\"os Problem about the Maximum Modulus of Littlewood Polynomials on the Unit Circl

Making a progress in an Erd"os problem dating back to 1957, we show that $$\underset{t\in \mathbb{R}}{\max }\mathrm{\mid }{P}_n(e^{it}){\mathrm{\mid }}^2\ge n+1+\frac{1}{38}{n}^{1\mathrm{/}3}$$ for every polynomial $P_n$ of degree $n$ with all of its coefficients in $\{-1,1\}$.